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Clarification about QTF proof regarding equality of QFT application and circuit application

I suppose there are two sources of confusion here -- one is the reversed order of the qubits between the circuit and the unitary, and the other is the use of $x$. In terms of qubit ordering, I think ...
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DFT like operation in the third step of Period finding and Discrete Logarithm algorithm

Your first question about making sense of $| \hat f(\ell_1, \ell_2)\rangle$ is at best vague. It is not clear what you want to know. Your second question is also unclear. What inverse are you talking ...
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How to intuitively interpret the QFT of a state?

The discrete Fourier transform acts on a vector $(x_0, ..., x_{N-1})$ and maps it to the vector $(y_0, ..., y_{N-1})$ according to the formula $$y_k = \frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}x_j\omega_N^{jk}...
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